Static stability index

The stability index has been implemented in the most recent form as described in detail by Reference Jamieson and JohnstonJamieson andJohnston (1998), which is based on the work of Reference RochRoch (1966) and Reference FöhnFöhn, 1987. The index is the fraction between the snow shear strength (critical snow shear stress) and the actual shear stress exerted either just by the snow itself or by the snow and a skier. The critical stress is determined by the Daniel’s strength plus a correction term that increases with increasing snow loading (Reference Jamieson and JohnstonJamieson and Johnston, 1998). Values of the Daniel’s strength are parameterized following Jamieson and Johnston (2001). The parameterization is based on density and grain type only and therefore makes no use of the SNOWPACK microstructure quantities. Therefore, based on new field data (Reference Chalmers and JamiesonChalmers and Jamieson, 2001), we introduce a refined parameterization for buried surface hoar. The strength is given as a linear function of several snow parameters. The parameters and coefficients are summarized in Table 1. Table 1 gives the original values from Reference Chalmers and JamiesonChalmers and Jamieson (2001) and our re-analysis after eliminating the variable “minimum grain-size”. It is necessary to treat surface hoar separately because the density-dependent formulations for the other grain types would not work for thin surface-hoar layers, where only ambiguous density values can be determined. Furthermore, surface hoar is one of the most frequent failure layers, so the newly available data are highly welcome.

Deformation-rate stability index

The two static stability indices represent the strength-to-force ratio for a separated column of snow. They neglect that the strength of snow depends on the deformation rate. Therefore, an additional stability index is proposed, which is based on the deformation rate of the snow layers. This index still does not include the full physical basis of avalanche formation, which would involve the description of fracture propagation in a three-dimensionally varying snow cover and eventually the tension failure at the avalanche crown. However, the deformation-rate analysis is expected to describe the situation of new-snow avalanches as well as wet-snow avalanches better than the static stability indices. It also benefits from the microstructure treatment of snow in SNOWPACK (Reference Lehning, Bartelt, Brown, Fierz and SatyawaliLehning and others, 2002a) because the deformation rate is determined by the grain-and bond sizes directly.

For the strain-rate-dependent critical stress in the bonds (or necks) of the snow matrix, we use a simplified form of the Reference Nadreau and MichelNadreau and Michel (1986) criterion. Their criterion for failure has been tested for strain rates in ice ranging from 10^–6 to as high as 10^–3 s^–1. This translates into snow strain rates approximately an order of magnitude smaller, which covers the range of snow strain rates of interest. The failure criterion takes into account the effect of pressure on the melting point of ice. Let σ_c be the critical stress in the necks at which the necks will fail. The failure criterion is then:

In the equation, p_h is the (negative) hydrostatic pressure that induces melting, σ₀ is the brittle fracture stress of ice and ϕ is the strain-rate-dependent function which determines the shape of the failure surface. The parameters are determined according to Reference Nadreau and MichelNadreau and Michel (1986):

In these empirical parameterizations, T_C is the temperature of ice in °C and ɛ̇_n is the deformation rate in the bonds between the snow grains. The calculation of this deformation rate is developed in Reference Lehning, Bartelt, Brown, Fierz and SatyawaliLehning and others (2002a). The final deformation-rate stability index is then given by:

which is the ratio between the critical stress in the bonds or necks and the actual neck stress, σ_n. σ_n is related to the snow overburden as detailed in Reference Lehning, Bartelt, Brown, Fierz and SatyawaliLehning and others (2002a).

Discussion of preliminary stability results for the IMIS station Klosters Gatschiefer

SNOWPACK has been mostly validated against data collected at our experimental site Weissfluhjoch Versuchsfeld (Reference Lehning, Bartelt, Brown and FierzLehning and others, 2002b). In the past two winters, data have also been collected at IMIS stations in the vicinity, so we now choose different locations, starting with the station Klosters Gatschiefer at 2310 m altitude. The station has more influence from the north and is thought to be representative for the situation north of the Davos area. Figure 1 pre-sents the stability results for that station for January– February 2003 together with the grain type representation. The time marker is set at 7 February, when the avalanche danger increased to the level “high” (4, according to the European avalanche danger scale; see also Fig. 5) and many avalanches including large-scale ones were observed. The righthand side of the figure shows the respective stability grain-type profiles for that date. The figure has been produced from the SNOWPACK SN_GUI.

Fig. 1. Time development of the natural stability index and grain types for the IMIS station Klosters Gatschiefer inJanuary– February 2003. The time development is shown for each snow layer and is colour-coded. At the marker position, the corresponding profile is given at the righthand side of the graph. For stability, not only is the stability index profile given but also the minimum is marked by an arrow (blue for natural index). Two additional arrows give the locations of the two remaining stability indices (deformation rate in pink, and skier in green). The minimum index values and their height are also written below the profile graph. The number for the stability class (S_class: 5) is meaningless in this graph. The stability class estimation is not yet implemented.

The profile information presented in Figure 1 is for the natural stability index. On the right of the graph, minimum values and the location of these minima for the remaining two indices are also given to allow easy comparison.

For the critical situation of 7 February, all three indices give values 51 and indicate instability. The deformationrate and skier indices locate the instability at the surface below the most recent snowfall event. This corresponds to failure surfaces of observed avalanches and failure tests in the vicinity of the station.

However, the stability development of Figure 1 also shows features that are clearly inadequate. At present, the critical shear stress for both natural and skier indices is mainly a function of grain type and density. Furthermore, the stability index describes the situation for a separated column of snow and does not take into account three-dimensional effects. Therefore, the natural stability index detects the weakest point in the profile in the layer of faceted crystals above the melt–freeze crust at 20 cm above the soil surface. The skier stability index finds the correct location of minimum stability because it is confined to the first 120 cm from the snow surface. It agrees with the deformation-rate index location of maximum weakness. This tendency of the natural stability index to diagnose maximum weakness close to the ground appears throughout the winter. This is because the increase in snow load exceeds the increase in estimated snow strength, which is a non-linear function of snow density only.

Another erroneous behavior is exhibited by the way stability changes due to metamorphic development. Following the long period of cloudless and cold weather in January, previously rounded grains (pink) become more faceted (blue) in the center of the snow cover. The snowfall events at the end of January cause the grains to become more rounded again afterwards. The development causes the stability first to decrease significantly and then to increase again. This is clearly in contradiction with observations. In the following, we will explore heuristically how to improve on these deficiencies.

Suggestions to improve natural and skier stability indices

While the deformation-rate stability index takes advantage of the advanced treatment of microstructure in SNOW-PACK, the other two formulations are a function of the descriptive grain type and density only. Clearly, the bonding between grains could be used to reduce the error of increasing instability with depth of snow-layer burial. The bonds grow slowly during dry metamorphism and fast when melt processes become important (Reference Lehning, Bartelt, Brown, Fierz and SatyawaliLehning and others, 2002a). Nevertheless, the small changes in bond size during dry (kinetic growth or equi-temperature) metamorphism are expected to increase snow stability significantly. Therefore we explore the effect of scaling the natural and skier stability indices with the logarithm of bond size:

Here ∑_∞ is the shear (Daniel’s) strength, σ_zz is the normal load and σ_xz is the shear force per unit area. The function ϕ(∑_∞; σ_zz) is a correction term describing the effect of overload on shear strength (Reference RochRoch, 1966). The scaling term contains the scaling factor f(1) the bond radius r_b and the minimum and maximum bond radiuses r_{min b} and r_{max b} , respectively. Since the scaling will produce a new stability index with the range where S_n is the original index, the scaling factor can be used to determine the magnitude of the scaling. Figure 2 shows the result for f = 0.7.

Fig. 2. Time development of natural and deformation-rate stability index as in Figure 1. For the natural stability index, an overload correction for all grain types and a bond-size scaling have been introduced.

Reference Jamieson and JohnstonJamieson and Johnston (1998) review formulations for the normal-load correction ϕ(∑∞; σzz). In their Figure 1, all grain types appear to show a similar shear strength dependency on normal load. Nonetheless, they use the normal-load correction only for fresh snow, decomposed particles and rounded grains. This restriction causes the erroneous behavior discussed above, that stability will decrease due to metamorphism and then increase again. Using a normal-load correction for all snow types will also help to limit the decrease of stability when moving to larger depths in the snow profile. We therefore suggest using the normal-load correction, which they suggest for decomposed particles and rounded grains,

for all grain types, except for buried surface hoar because the new formulation of shear strength for surface hoar (Table 1) already contains an explicit dependency on normal load.

The two corrections discussed above are applied to both the natural and the skier stability indices. In the following we discuss results obtained with the adapted indices.

Results with adapted formulations

Figure 2 presents the development of the natural and deformation-rate stability indices. The deformation-rate index has not been changed. The bond-size scaling together with the introduction of the overload correction for all grain types (except surface hoar) has the desired effect that the natural stability index no longer shows the destabilization zone at the end of January in the middle of the snow cover due to kinetic metamorphism. Also, the tendency to show increasing instability for deeply buried snow layers is now reduced. For the critical situation of 7 February, all three stability indices indicate instability for the fresh snow layers.

Analyzing the behavior of the deformation-rate index, it becomes obvious that it mostly has minimum values at relatively recent snow layers close to the surface. The index is less specific than the more empirical static stability indices. For example, layers of buried surface hoar are not necessarily captured by this formulation. The index is constructed to pick up dynamic layers, where changes in the snow microstructure are taking place. The index correctly predicts that old and settled layers of snow are no longer dangerous. An example of its usefulness in new-snow situations is given in Figure 3. The deformation-rate stability index is shown, together with the temperature development for December, at Klosters Gatschiefer. One of the most difficult judgments for avalanche-warning experts is to what extent a sudden increase in temperature can destabilize the snow cover and lead to avalanches. In Figure 3 it can be seen that warming creates a distinct layer of high deformation at approximately 20 cm below the snow surface. This development starts following the warming period on 18/19 December. The following cooling phase makes the layer less active. With the snowfall of 23 December and the increase in temperature at the same time, the layer becomes active again. In this area, spontaneous slab avalanching has been observed following the snowfall and the warming. The failure layer corresponds to the layer found by the index, which gives an intermediate stability value of 1.3. Note that at the station Klosters Madrisa, on the opposite side of the valley, a similar development is observed (not shown).

Fig. 3. Time development of deformation-rate stability index and temperature. Rising temperatures create a weak layer below the surface. On 23 December, avalanching has been observed in the area.

In the previous subsection, surface hoar was mentioned as a possible problematic layer. In winter 2002/03, significant surface hoar was formed during a long period of cloudless weather between 5 and 16 December. This surface hoar was subsequently buried and was the primary weak layer for skier-triggered avalanches approximately until the snowfall events at the end of the first week of January. Figure 4 explores the skier stability index for this situation. The skier stability index gives a result almost identical to that given by the natural stability index (not shown). The profile drawn for 20 December shows that the buried surface hoar is the primary instability, even though the absolute value of the index, 1.4, is not very low. Two more local instabilities due to buried surface hoar are found in the profile below the primary instability. Note that these three surface-hoar instabilities are no longer present on 7 February (Fig. 2). In fact, the index shows a rapid increase of stability for the surface hoar (cf. also Fig. 5). By the beginning of January, the numerical value of the stability index has increased from 1.4 to 2. However, the buried surface-hoar layer remains a local instability in the profile (not shown).

Fig. 4. Time development of skier stability index for December–January for Klosters Gatschiefer.The surface hoar formed during the period of fair weather in December creates a local instability when buried by fresh snow.

Fig. 5. Time series of minimum values of skier and natural stability indices in comparison with development of forecasted avalanche danger level for Klosters Gatschiefer (y axis inverted). Parts of the time series show a good correlation.

The most interesting two situations of winter 2002/03 have been analyzed and an example of the effect of a sudden warming on the deformation-rate index has been discussed. Figure 5 gives the time-series representation of the minimum values of the skier and natural stability indices for the IMIS station Klosters. For comparison, the avalanche danger level valid for the area is plotted on the same graph. Note that the degree of danger is plotted on an inverse axis to allow a better visual comparison to the stability index. Ideally, stability indices and the danger-level curve should be correlated. The visual inspection suggests a remarkable correlation. As discussed above, the most dangerous situations are recognized by our stability model. Other local minima of the stability index are not correlated with an increase in the danger level. For example, for the snowfall events starting at the end of January, a destabilization is predicted by the indices, while no corresponding general increase of the danger level was issued.

As a further example, the time series of stability indices at the IMIS station Davos Hanengretji are shown in Figure 6. The instability on 7 February is also clearly recognized at this station. The burial of the surface hoar layer produces a larger instability than at Klosters Gatschiefer. Also, this instability is slower to stabilize.

Fig. 6. Time series of minimum values of skier and natural stability indices for the IMIS stationDavosHahnengretji.The surface hoar layer produces a larger instability than for Klosters Gatschiefer.

As discussed in the introduction, it cannot be expected that the danger level is closely related to the results of our stability index for individual IMIS stations. Therefore, we consider the stability results for four additional IMIS stations (Davos Bärentälli, Davos Parsenn, Weissfluhjoch Versuchsfeld and Klosters Madrisa) in the vicinity. All four stations give a clear indication of instability on 7 February. Only one of these four (Davos Bärentälli) indicates instability for the buried surface hoar. The third instability found at Klosters Gatschiefer, related to the snowfall events at the end of January, is not present at any of the other stations.

These results indicate that the true situation is better represented by interpreting the results at all available stations.